\section{1.21. Coherence } 
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Why $\mathcal{O}_X$ is a {\color{red}coherent} sheaf of rings for smooth complex algebraic variety $X$?
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To understand why the structure sheaf \(\mathcal{O}_X\) is a {\color{red}coherent} sheaf of rings on a smooth complex algebraic variety \(X\), we need to recall the definitions and key properties from algebraic geometry.

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1. Definitions. A {\color{red}coherent sheaf} \(\mathcal{F}\) of \(\mathcal{O}_X\)-modules on a ringed space \((X, \mathcal{O}_X)\) is a sheaf such that:

(i) \(\mathcal{F}\) is of {\color{red}finite type}: every point \(x \in X\) has an open neighborhood \(U\) such that there is a surjection \(\mathcal{O}_U^{\oplus n} \twoheadrightarrow \mathcal{F}|_U\).
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(ii) For any open \(U \subset X\) and any morphism \(\phi: \mathcal{O}_U^{\oplus m} \to \mathcal{F}|_U\), the kernel \(\ker(\phi)\) is also of {\color{red}finite type}.

A {\color{red}coherent sheaf} of rings usually means a sheaf of rings \(\mathcal{A}\) that is {\color{red}coherent} as a module over itself. So, to say \(\mathcal{O}_X\) is a {\color{red}coherent} sheaf of rings means: \(\mathcal{O}_X\) is {\color{red}coherent} as an \(\mathcal{O}_X\)-module.


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2. Key Fact: Coherence of \(\mathcal{O}_X\) on Smooth Varieties

Let \(X\) be a smooth complex algebraic variety (i.e., a smooth separated scheme of finite type over \(\mathbb{C}\)). Then \(\mathcal{O}_X\) is a {\color{red}coherent} sheaf of rings. 

This is a consequence of the following classical result:

Theorem (Oka–Cartan–Serre): On a complex manifold (or smooth complex algebraic variety), the structure sheaf \(\mathcal{O}_X\) is {\color{red}coherent}.

But let's explain this algebraically.


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3. Algebraic Reasoning

A smooth complex algebraic variety \(X\) is {\color{red}locally of finite type} over \(\mathbb{C}\) and regular (since smooth over a field implies regular).

Locally, \(X\) is covered by open affines \(U = \operatorname{Spec} A\), where \(A\) is a finitely generated regular \(\mathbb{C}\)-algebra.

Regular rings that are finitely generated over a field are Noetherian (Hilbert's basis theorem) and regular $\Rightarrow$ locally factorial $\Rightarrow$ Cohen-Macaulay, but more importantly:
  
If \(A\) is a Noetherian ring, then the structure sheaf \(\mathcal{O}_{\operatorname{Spec} A}\) is a {\color{red}coherent} sheaf of rings.

Why? Because on an affine Noetherian scheme, quasi-coherent sheaves correspond to \(A\)-modules.

The sheaf \(\mathcal{O}_X\) corresponds to the module \(A\).

Since \(A\) is Noetherian, submodules of finitely generated \(A\)-modules are finitely generated.

This implies that \(\mathcal{O}_X\) satisfies the definition of a {\color{red}coherent} sheaf: any map \(\mathcal{O}_U^n \to \mathcal{O}_U\) has a finitely generated kernel (because it corresponds to a map \(A^n \to A\), whose kernel is a finitely generated \(A\)-module).

Thus, on any Noetherian scheme, the structure sheaf \(\mathcal{O}_X\) is {\color{red}coherent} if and only if the local rings \(\mathcal{O}_{X,x}\) are {\color{red}coherent} rings (i.e., finitely generated ideals are finitely presented). 

But every Noetherian ring is {\color{red}coherent}. Therefore, since a smooth complex algebraic variety is locally Noetherian (in fact, of finite type over \(\mathbb{C}\), hence Noetherian), it follows that $\mathcal{O}_X$ is a {\color{red}coherent} sheaf of rings on $X$.


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4. Remark on Analytic vs Algebraic

In the analytic category (complex manifolds with holomorphic functions), the coherence of \(\mathcal{O}_X\) is a deep theorem (Oka's coherence theorem).

In the algebraic category, over a Noetherian base (like \(\mathbb{C}\)), coherence of \(\mathcal{O}_X\) is much easier: it follows directly from the Noetherian property.

Since a smooth complex algebraic variety is an algebraic object of finite type over \(\mathbb{C}\), we are in the algebraic (Noetherian) setting, so coherence is automatic.


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Final Answer:

Because a smooth complex algebraic variety \(X\) is locally Noetherian (being of finite type over \(\mathbb{C}\)), and the structure sheaf \(\mathcal{O}_X\) on any locally Noetherian scheme is {\color{red}coherent} (since Noetherian rings are {\color{red}coherent}), it follows that \(\mathcal{O}_X\) is a {\color{red}coherent} sheaf of rings.


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